91Ó°ÊÓ

Use the Product Rule to differentiate the function. \(g(x)=\sqrt{x} \sin x\)

Consider the linear function \(y=a x+b .\) If \(x\) changes at a constant rate, does \(y\) change at a constant rate? If so, does it change at the same rate as \(x ?\) Explain.

Find the rate of change of the distance between the origin and a moving point on the graph of \(y=x^{2}+1\) if \(d x / d t=2\) centimeters per second.

Find the rate of change of the distance between the origin and a moving point on the graph of \(y=\sin x\) if \(d x / d t=2\) centimeters per second.

Let \(A\) be the area of a circle of radius \(r\) that is changing with respect to time. If \(d r / d t\) is constant, is \(d A / d t\) constant? Explain.

The included angle of the two sides of constant equal length \(s\) of an isosceles triangle is \(\theta\). (a) Show that the area of the triangle is given by \(A=\frac{1}{2} s^{2} \sin \theta\). (b) If \(\theta\) is increasing at the rate of \(\frac{1}{2}\) radian per minute, find the rates of change of the area when \(\theta=\pi / 6\) and \(\theta=\pi / 3\). (c) Explain why the rate of change of the area of the triangle is not constant even though \(d \theta / d t\) is constant.

The radius \(r\) of a sphere is increasing at a rate of 2 inches per minute. (a) Find the rate of change of the volume when \(r=6\) inches and \(r=24\) inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though \(d r / d t\) is constant.

All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.